Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, naturals)

On 12/17/2024 02:03 AM, FromTheRafters wrote:

Ross Finlayson wrote :

On 12/16/2024 03:56 AM, joes wrote:

Am Mon, 16 Dec 2024 09:27:19 +0100 schrieb WM:

On 15.12.2024 21:20, joes wrote:
>

Duh. All naturals are finite. You need to actually remove all inf.many
of them.

That is not possible with definable naturals:

Duh again. No natural is infinite.
>

And numbers which succeed ∀k ∈ ℕ

There is no natural larger than all others.
>

>
Oh, one of my podcasts next week will have
"natural infinities", because there's no
standard model of integers, only fragments
or extensions, making that there _are_ natural
models with infinite members.
>
It's a simple consequence of comprehension
and quantification, in theories that don't
define it away.
>
How could there be a natural integer smaller
than all others? Of course you may know
that "1" is a very large cardinal, then
as with regards to whether "0" exists at all,
whether it's finite, and otherwise via
comprehension and quantification,
"natural zeros".

>
And in another area of mathematics we are not talking about, one can
equal zero.

Ah, that's always erroneous, "x =/= x" or "x = !x", with
regards to identity, tautology, equality, those being
relayed as intensionality (identical) and extensionality (equivalent),
that "ex falso quodlibet" is _not_ so, only "ex falso nihilum".
So, from having arrived at "this sentence is not true", a
merest example of a non-commital otherwise "Liar paradox",
then we don't have any validation of "material implication",
so that there's no "ex falso quodlibet", only "that's wrong".
Then, when what you got is an apparent contradiction, must
mean that there are two different theories, then as with
regards to _one_ theory that explains both while disambiguating
each, is for a fuller, more thorough, reasoned, dialectical
account, because reason wins, and "ex falso quodlibet"
is not a thing.
Then, when "x =/= x" or "x = !x" are arrived at,
in some respects these make for the reasoning of
the alternation or vacillation of terms, when they do.
Then, where the soft-ball straw-man (a fallacious reasoner
or "irrational reasoner") has that for these things,
that notions like
infinite-middle
make for that there aren't standard models of integers,
as it's arrived at the Russell's retro-thesis is hypocritical,
here just makes for systems where cardinality is relevant
and systems where it isn't, with regards otherwise
to wider concerns.
So, "where one can equal zero meaning something so is not so",
is not mathematics at all, it's contingency pending failure
in reasoning.